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We study Brezis-Nirenberg type problems, governed by the double phaseoperator $- \mathrm{div}\left(|\nabla u|^{p-2}\, \nabla u + a(x)\, |\nablau|^{q-2}\, \nabla u\right)$, that involve a critical nonlinearity of the form$|u|^{p^\ast - 2}\, u + b(x)\, |u|^{q^\ast - 2}\, u$. Both for the local caseand for related nonlocal Kirchhoff type problems, we prove new compactness andexistence results using variational methods in suitable Musielak-Orlicz Sobolevspaces. For these functional spaces, we prove some continuous and compactembeddings that are of independent interest. The study of the local problem iscomplemented by some nonexistence results of Poho\v{z}aev type.

Critical growth double phase problems: the local case and a Kirchhoff type case